The analysis of the efficiency and efficacy of supporting structures requires an understanding of the elements of strength of the materials involved. Thus, when a column is subjected to an eccentrically applied axial load, the column will bend in response to the degree and direction of eccentricity. In addition, once bending is initiated further bending takes place in the plane defined by the centroid of the column and the point of application of the load; the resistance of the column to bending in that plane ultimately determines the load bearing capacity of the column.
If a vertically disposed column is a cylinder or a tube, and the force is applied concentrically, the resistance of the column to bending will be equal in all directions because the material is placed uniformly around the axis and no bending below maximum load (critical stress) takes place. If the column, however, has an irregular cross-section, then the column will fail by bending on its axis of least resistance; i.e. the principal axis of the section about which its radius of gyration is least.
When a column bends, the fibers in the material of the column which resist the bending are subjected to tension on the outside of the bend and compression on the inside, between which lies a "neutral axis" running from end to end of the column where there is neither tension nor compression. The resistance to bending is provided by all the fibers which are not on the neutral axis and since the fibers which are farthest from the neutral surface are subjected to the greatest amount of deformation during bending, they contribute the most to the resistance. The relative resistance to bending can be determined by calculating the radius of gyration of the structure in question for its various bending axes*. Thus, the weakest direction of bending of a column (i.e. the axis of least resistance to bending), is the direction in which the radius of gyration is the least. FNT *(More specifically, the radius of gyration r conforms to the following formula: ##EQU1## FNT where I is the amount of inertia, A is the area).
Accordingly, hollow tubes make the best columns because they provide a maximum radius of gyration in all directions. Although hollow rectangular shapes are somewhat weaker than tubes, all other things being equal, they also make good columns. They have certain practical advantages over tubes in that they are easier to attach to other structures and they are usually cheaper. An important requirement for all hollow shapes, however, is to have a continuous, unbroken surface. Thus, a tube or other hollow shape which has an open seam in its surface is very much weaker than if the surface is integral and continuous. It follows that I, H or channel cross-sections are weaker than enclosed hollows for essentially the same reason. The free edge of a flange has nothing other than its own rigidity to keep it from buckling. Of course, even though I, H and U cross-sections are weaker than tubes and hollows, they are often used in preference thereto because they are cheaper and more readily available. Also, I, H and U shapes can be easily incorporated into other structures which combine with them to give at least in part the effect of a hollow structure. Even less desirable for free standing, load bearing columns, struts, or braces, are simple angle irons which traditionally comprise a pair of flanges disposed at right angles to each other, often with one of the flanges being wider than the other. Usually, several angle irons will be combined to form a composite structure in which the maximum radius of gyration of one angle iron is aligned so as to reinforce the least radius of gyration of an adjacent angle iron and thereby to attain, at least in part, some of the advantages of the more complicated enclosed cross-sections. Another disadvantage of the traditional angle iron is that the centerline of the column (the centroid) lies between the two flanges. Thus, if the load is to be applied in the optimal manner, i.e. with the thrust axis on the centroid, a bridging connection to both flanges is required. Otherwise, simply connecting the load to one flange of the column subjects the column to an undesirable bending stress arising from the eccentricity of the connection. In addition, simple angles tend to twist under loading and this, in turn, complicates the predictability of their maximum loading condition.
Due to the low cost and ready availability of the traditional angle iron, however, as well as the relative ease with which simple right angled cross-sections can be cross-braced or combined with other elements to form composite structures in which the various columns or braces reinforce each other, such conventional angle irons have been widely used as columns in supporting structures.
A typical illustration of such a useage has been in telescoping grandstand seating structures adapted to be pulled from a stowed, telescoped position (usually against a wall) to a fully extended position set up for use. In such structures, the individual tiers of seats are supported independently on vertical columns which are fitted with rollers at their bases on which the tires are rolled out or in between the stowed and set-up positions. When the tiers are rolled into the stowed position, each lower tier together with its entire supporting columns and braces must nest within the supporting columns and braces of next above tier. Normally, hollow rectangular cross-sections are employed for the side columns at the outer ends of the tiers, and the center portions of the tiers are supported by angle iron columns connected diagonally between the bases of the side columns and the center portions of the tiers, the diagonal disposition of the angle irons being dictated by the need for bracing the stand against lateral swaying. The use of angle irons is also desirable in the specific context because angle irons can nest conveniently and it is desirable to use them so that telescoping the tiers can be done within the smallest space possible. On the other hand, in order to achieve efficient nesting, the connection of the load to only one flange of the column is desirable. This, of course, subjects the column to bending stress due to eccentric application of the load and requires the use of larger and heavier angle irons. Still another factor requiring the use of larger and heavier angle irons has to do with the "slenderness ratio" of the column. Obviously as the length of an unsupported column of a given cross-section is increased, its resistance to bending and hence its maximum load bearing capacity decreases. The slenderness ratio is a factor which is used to judge the efficiency of unsupported columns, and to indicate when a given cross-section has reached its maximum safe length. The slenderness ratio is determined by dividing the length of the column by the least radius of gyration of the column. The American Institute of Steel Construction has established a slenderness ratio of 200 as maximum for unsupported columns, struts or braces, and this requires the use of substantially larger and heavier angle irons for the long, compression members needed to support the upper tiers of the grandstand seating structures described.
The present invention stems from the discovery that the conventional angle iron, when used as an unsupported column, makes an inefficient use of the metal. For example, with an angle iron having 1" flanges and a thickness of 12 gauge (0.1045") the maximum radius of gyration is 0.803, whereas the minimum radius of gyration is less than half of that, i.e. 0.392. The column, of course, has no greater resistance to bending than the resistance thereto on its weakest axis, but the fact that on other axes, the column has much greater resistance reveals that the column is not making use of the full potential of its metal.
Accordingly, a basic object of the present invention is to provide a more efficient column which employs the advantages of the conventional angle iron, of simplicity, cost, ready producibility, availability, and ease of incorporation into other structures, which also uses the material of the cross-section more efficiently so as to provide less expensive and lighter columns than conventional angle irons having the same load bearing capacity. Another object is to provide such a column which is suitable for use as a compression bearing, unsupported, pin connected column, brace or strut. A further object is to provide such a column meeting the foregoing objects which is also suitable for nesting together with a multiplicity of other, like columns, and to which the load may be connected at only one flange with a significant reduction in bending stress due to eccentricity of connection as compared to conventional angle irons.